Muon Acceleration in Cosmic Ray
Sources
Spencer Klein, LBNL & UC Berkeley
In collaboration with Julia Becker Tjus, Rune Mikkelsen
and Walter Winter
Introduction to particle acceleration in cosmic sources
Muon Acceleration
Neutrino Flux Enhancement
Some Comments on Source Models
Plasma Wakefield Acceleration
Magnetars
GRBs
Conclusions
Refs.: Ap. J. 779, 106 (2013) & Astron & Astrophy. 569, A58 (2014)
Motivation: transient sources
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Many of the most interesting astrophysical phenomena are
transient and/or variable
 Gamma-ray bursts
Supermassive star collapse
 Colliding black holes
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 Supernovae
 Active
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galactic nuclei are exhibit variability at diverse
energies
Short time scales require high accelerating gradients
Gradient g > Emax/ct = 1020 eV/cτ
 τ=100 s (GRB...) -> g > 3.3 GeV/m [laser-plasma in lab]
 τ=1 day (AGN..) -> g > 4 MeV/m [SLAC/4]
 If the source is a relativistic jet moving toward Earth, time
dilation reduces the gradients by Γ2 (Γ= source boost)
Cosmic acceleration mechanisms
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Acceleration likely occurs in an energetic, turbulent plasma
containing ionized gases and magnetic fields
Astrophysical plasma may contain shock fronts, collisions
between clouds of plasma moving at different speeds.
In Fermi acceleration, when a charged particle encounters a
shock front moving toward it, it may rebound gaining energy.
 Multiple encounters needed to reach high energies
Alternative models
 Astrophysical plasma wakefield accelerators allow for very
high gradients in multiple types of sources
 In some magnetar models, acceleration is via a strong
electric field
Fermi shock acceleration
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Type 1 acceleration involves encounters
between single particles and moving
shock fronts.
 Energy gain ζ ~ 4/3 ∆β, where ∆β is
the velocity difference between the
upstream and downstream media
Type 2 acceleration involves encounters
between single particles and randomly
oriented plasma blobs
 ζ ~ 4/3 β2– slower than type 1
ζ <2 requires many encounters to reach
high energies
After each encounter, there is a probability Pesc of
escape or another encounter
 Leads to a power law spectrum, dN/dE~ E-~2
More detailed modelling
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Detailed models exist for most types of source
Two different classes of geometries
 Spherical sources
Supernova remnants…
 Magnetic fields provide confinement, leading to
repeated particle-shock front encounters
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 Relativistic
particle jets
Emitted from axes of active galactic nuclei, GRBs,
 Linear accelerators with many shock fronts
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Many sources are transient – GRBs, magnetars
 Even supernova remnants have short
(~1,000 y) lifetimes
Most models predict a spectrum dN/dE~E-~2
 Spectrum softens to E-2.7-3.0 en-route to Earth
Other acceleration mechanisms

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In plasma wakefield acceleration, periodic
variations in charge density (i.e. plasma
waves) lead to very high accelerating
gradients
 Similar to more conventional accelerators
 Astrophysical gradients of 1014-1016
eV/cm quoted in papers
In some magnetar models (Arons, 2003),
particle acceleration is from particles ‘surfriding’ the expanding electromagnetic fields
 Magnetars are fresh neutron stars with
ultra-high (Peta-Gauss) surface magnetic
fields.
 dN/dE ~ E-1
ν and γ production in hadronic accelerators

γ and ν are produced from π & K ion decay when
accelerated nucleon interact with beam-gas or
photons
Most interactions in accelerator; in-flight
interactions also occur.
 These interactions may not be in the same region
as the acceleration
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The ratio of π0 γ to π ± -> µ ± νµ  eνµνe is fixed
by the hadron physics

Leads to a 2:1 νµ:νe ratio
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Oscillation leads to 1:1:1 νµ:νe: ντ at Earth
• Experiments have little at-source flavor sensitivity
Exception – heavy quarks produce mostly ν, few γ
ν energy spectrum is similar to that of CR nuclei
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ν flux estimates
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Relate to photon flux measured at GeV energies, from
Fermi & Cherenkov telescopes
 Extrapolation in energy
 Photons may be absorbed
 Photons may come from electromagnetic interactions
Relate to cosmic-ray nucleon flux
 ν flux depends on beam-gas/photon density in source
 Maximum Eν is ~ 5% of maximum CR nucleon energy
Maximum total Eν:ECR ratio is ~ 1:1
 The Waxman-Bahcall bound
 If ratio were higher, sources would be ‘choked,’ without
visible cosmic-ray emission
The observed IceCube flux is close to this bound
Muon acceleration and energy loss
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2/3 of the neutrinos come from µ decay
If the muons gain or lose energy during their
lifetime, the ν flux will be enhanced/reduced
Energy loss can occur via bremsstrahlung,
pair production, photonuclear interactions, or
synchrotron radiation
 Synch. Rad is usually most important
A similar phenomena occurs for pions/kaons,
except that the lifetimes are 100 times
shorter
Kashti & Waxman (2005)
µ acceleration
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For small accelerating gradients g (in keV/cm), energy
gain/loss for an individual muon that lives lifetime t is
As long as ∆Eµ < Eµ,
∆Eµ = gγct/mµ
 Lorentz boost lengthens time acceleration occurs
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If the gradients are larger, then
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Efinal = Einit exp(gct/mµ)
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For g> mµ/ct ∼ g0 = 1.6 keV/cm, acceleration is large
Similar considerations apply to π
Since τπ ~ 0.01 τµ, π acceleration is much less important
 Included in our calculations
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µ Acceleration with energy loss
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If g> mµ/cτµ muons gain significant energy before they decay.
They also lose energy
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µ-matter interactions
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µ-photon interactions
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Usually smaller than p-photon energy loss
Synchrotron radiation
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Much smaller than for p-matter energy loss
∆E/∆t ~ γ2B2
The only energy loss mechanism that is larger for µ than protons. So, we
focus on it, as the limiting case
Maximum energy Ec when
dEgain /dx= dEloss/dx
Calculating the ν spectrum
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Start with a proton spectrum with a given index
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Generate π with the same index
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Pion gets ~ 20% of proton energy
Propagate π, including acceleration and energy loss
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Decay times are distributed exponentially
Assume νµ from π decay takes ¼ of energy
 Propagate µ, including acceleration and energy loss
 Divide µ energy equally among e, νµ, ν e

longest lived π make the biggest contribution to the
spectrum
 The
Resulting ν flux
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ν flux is greatly enhanced, up to Ec, where energy loss by
synchrotron radiation balances the energy gain
At high gradients g >> g0, ν spectrum hardens significantly, to
roughly N ~ E -g0/g , not very dependent on the initial spectrum
 Maximum energy determined by accelerator length, or when
Egain = Eloss
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Enhancement
Enhancement
Enhancements for an E-2 spectrum at 0.5 keV/cm & 5 keV/cm gradients
Limits on g and opacity
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Waxman & Bahcall used the measured 1019-20 eV
cosmic-ray flux and assumed the maximum opacity Ο=1
to set a limit on the neutrino flux.
 Reverse to set limits on opacity Ο < φobserved/φWB
IceCube prefers a spectrum softer than dN/dEn ~ E-2, but
not decisively so. We will consider E-2 here.
These calculations used the IC40 limit on diffuse νµ
 The IceCube 3-year contained event flux is close to the IC40
limit, so we can just replace ‘limit’ with measurement
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The data can be used to set 2-dimensional limits on
opacity (or density) and accelerating gradient
 Muon acceleration alters the spectrum.
 Calculate the number of events needed to be seen as
an excess using new spectrum, based on the
published below-the-horizon effective area vs. Eν
Ο max max vs. g
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1-d limit, Ο ~ 0.4 holds up to about 500 eV/cm
At higher gradients, maximum opacity drops rapidly
 Magnetic field enters via changing Ec
For g > ~ 8 keV/cm opacity < 10-7
 A rather tight constraint
Matter density (ρ) constraints
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Probability of interaction is O = σρL
 L = 1020 eV/g
 For g > 5 keV/cm, density ρ < 106/cm3
Problematic for some accelerator models
Plasma Wakefield Acceleration
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Allows very high accelerating gradients
 1 GeV in 10 cm observed in laboratory
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(10 GeV in 1 meter this fall?)
Proposed for diverse astrophysical sources
 E. g 1013 keV/cm in GRBs
 AGNs?
PWA uses an excited plasma for acceleration
 In cosmos, ‘magnetowaves’ excite plasma
 Computer simulations show that this requires
density ~ 1010/cm3
 This density is ruled out by the neutrino flux
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Can a PWA operate at significantly lower densities?
P. Chen et al., 2002, F. Y. Chang et al. 2008, 2009…
Magnetars
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Newborn neutron stars with
Peta-Gauss fields
Very high electric fields can develop
in (nearly) magnetic field free regions
 Region size (~30,000 km) requires high accelerating
gradients, > 3*107 keV/cm
 These regions are also short-lived
IC40 ν flux limits require that the matter density in
these regions be very low, < 2*106/cm3
 Is this realistic in the immediate neighborhood of a
post-collapse environment?
J. Arons, 2003
Caveats
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Assumes that ν-producing interactions occur at the
acceleration site, rather than after the acceleration.
 The matter is in the accelerator.
Assumes (~) that acceleration occurs linearly.
 Stochastic acceleration OK as long as there are multiple
stochastic encounters/particle.
 Varying gradients during acceleration may change the
magnitude of the enhancement.
We focus on energy loss due to synchrotron radiation.
If muon acceleration is large enough, it will drain the
accelerator.
 Don’t take the exact magnitudes of the enormous
enhancements too seriously.
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They do show that something big is going on, though.
GRBs
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Short bursts (<2 s or > 2s) imply short
acceleration times
Here take an average t = 10 s burst length
 g > 3*104 keV/cm for 1020 eV
Assume that the accelerator (fireball) is
moving toward us at boost Γ
 Accelerator length increases by Γ
 Maximum energy decreases by Γ
 g decreases by Γ2
 For Γ = 100, g = 3 keV/cm > 1.6 keV/cm
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µ acceleration enhances the neutrino flux
greatly
Simplified GRB ν energy spectrum
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Broken power law with 2 energy breaks
 Central region corresponds to proton spectrum (E-2)
 Lower energy region is below pion production
threshold in jet frame
 Above the higher energy break, secondary particles
(π,K) lose energy before decaying
Waxman & Bahcall, 1997, Hummer Baerwald & Winter, 2012
GRB modelling
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µ acceleration has been considered in a more
detailed GRB model
Acceleration occurs at the collision of expanding
two shells
 Acceleration at shock front boundaries, radiation
in downstream plasma
GRB modelling
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Individual GRBs are modelled, using measured characteristics
 Luminosity, fluence, duration, redshift, & observed
spectrum
 From these, internal parameters are inferred, and used in a
detailed model of transport and acceleration
Four representative GRBs are used as examples
 ‘SB’ = ‘standard burst’
Time scales
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The effect of muon, pion and kaon acceleration
depends on the relative time scales for acceleration vs.
interaction and/or decay
 Species are treated via in coupled differential
equations
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Different diffusion, energy loss, etc.
Diffusion models
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Alters Pescape – probability of reacceleration
Kolmogorov and Bohm diffusion have different energy
dependencies
Calculation done for steady state sources
 Large concentration abundance at critical energies,
where energy gain = energy loss
µ,π,K flux
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Steady state densities
ν flux
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ν flux enhancement varies
with model
Enhancement by factors of
2-10 for E> 1016 eV
Enhancements largest in
GRBs with jets with high
Lorentz boosts and low
magnetic fields (to minimize
synch.rad.)
ν flavor ratios
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µ acceleration or energy loss alters the flavor ratio from the
usual (for π/K decay) νe:νµ:ντ=1:2:0
Can get distinctive flavor ratio variation with energy
Plot from W. Winter, 2014
Flavor ratios on Earth
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Flavor ‘triangle’ collapses to almost a line
 Slight triangle width since θ23 ~ π/4, θ13>~ 0
Anything outside this triangle is beyond the standard model
Small observable differences for very different at-source
compositions
Plots from Gary Binder, 2014
Conclusions
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The inclusion of π/µ acceleration greatly increases the
predicted ν flux in models with high accelerating gradients.
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This ‘breaks’ the standard relationship between γ and ν
fluxes.
One can use the IceCube data limit to set 2-dimensional
limits on opacity (or density) and accelerating gradient.
For compact short-duration sources, these limits are quite
constraining.
 These limits rule out published models invoking plasma wave
acceleration.
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A detailed calculation has been done for GRBs; the ν flux
is enhanced by a factor of 2-10 at energies above 1016 eV.